# Indian Statistical Institute, ISI BStat & BMath 2021 UGB Solutions & Discussions

## Problem 1.

There are three cities each of which has exactly the same number of citizens, say . Every. citizen in each city has exactly a total of friends in the other two cities. Show that there exist three people, one from each city, such that they are friends. We assume that friendship is mutual (that is, a symmetric relation).

# Solution:

From the citizens, take one who has a maximum number of acquaintances from one of the two other cities. Suppose it is citizen from the first city, who knows citizens from the second city. Then knows citizens from the third city, since . Consider citizen from the third city, who knows . If knows at least one citizen from the acquaintances of in the second city, then is a triple of mutual acquaintances. But if knows none of the acquaintances of in the second city, then, in this city he does not know more than citizens, and hence, in the first city, he does not know less than citizens, which contradicts the choice of .

## Problem 2.

Let be a function satisfying . Assume also that satisfies equations (A) and (B) below.

for all integers .
(i) Determine explicitly the set .
(ii) Assuming that there is a non-zero integer such that , prove that the set is infinite.

# Solution:

Let in (A):

Since we deduce

Let in (B):

Either or

The function takes only two values and (both values are really in the range of , because ).

Assume that there exist a non-zero such that Then Take in (A):

It follows that hence and So, Further, from (A) we find that if then

By induction, for all and the set
is infinite.

## Problem 3.

Prove that every positive rational number can be expressed uniquely as a finite sum of the form

where are integers such that for all .

# Solution:

Let be a positive rational number. Define so that

Assume that are defined in such a way that

Define

Since

we necessarily have So, there exists a sequence of integers such that

and for all

For large enough we have so

The latter is only possible when

Now assume that there are two representations

with integer coefficients such that for Let be the first index such that By symmetry we can assume that Then

Consider inequalities

Hence, the equality is impossible.

## Problem 4.

Let be a differentiable function whose derivative is continuous, and such that for all If is not the identity function, prove that must be strictly decreasing.

# Solution:

We differentiate the relation

for all hence preserves the sign on Assume that for all Then is strictly increasing. For some we have (as is not the identity function). If then

but this also contradicts So, the assumption leads to a contradiction. Then for all and is strictly decreasing.

Let and

If for all , and

then find

# Solution:

The relation implies that

All coefficients that correspond to odd indices are equal to zero. So,

Consider polynomial By assumption,

Observe that is the coefficient near in roots of the polynomial are known:

In particular, and

Then and

## Problem 6.

If a given equilateral triangle of side length lies in the union of five equilateral triangles of side length , show that there exist four equilateral triangles of side length whose union contains .

# Solution:

We show that necessarily Assume that Consider an equilateral triangle of side length The distance between any two points in is Now consider an equilateral triangle of side length Consider the following configuration:

There are six points in the triangle with pairwise distances Hence, if then one needs at least six equilateral triangles of side length to cover an equilateral triangle of side length So, and the configuration above shows that equilateral triangles of side length are enough to cover an equilateral triangle of side length

## Problem 7.

Let be three real numbers which are roots of a cubic polynomial, and satisfy and Suppose Show that

# Solution 1:

We have

The derivative of

The polynomial is increasing on decreasing on increasing on Then necessarily

Further,

i.e. and and

Finally,

and

## Problem 8.

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is . The square at the bottom has side length and the top square has side length Water is filled in at a rate of cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?

# Solution :

Denote by the volume of the water after hours the water started to fill the pond. Then (cubic meters). Let be the level of the water and be the side of the square at the top of the water after hours the water started to fill the pond.

Extend the truncated pyramid to get the non-truncated pyramid and let be the height of the added part. Then we have the following picture

Similarity of triangles implies relations

Multiplying them we find

Hence,

Now,

Differentiate this:

Then

## Video Solution:

• Abhinav

In two questions I messed up in the last line where we finally write the answer .
Like instead of 19/27 or 19/3(3)^2 in the last question , is wrote 19/3(3) forgot the square , other wise all was fine until the last line …..
And I’m polynomial question , I wrote 1-(9!)^2 instead of plus sign in last again , how much marks would that cost , can I get a 9/10 probably ,so email me

• Shubhrojyoti Dhara

I feel sir made a mistake out there because the constant term of Q(x) is indeed -(9!)^2 as putting 0 in (x-1)(x-4)(x-9)……(x-81)….would give us (-1)(-4)(-9)….(-81) = -(9!)^2

• Yes, there were some calculation errors. Thanks for notifying me. Corrected.

• But sir the polynomial should be x^2(x^2 - 1^2)(x^2-2^2)...(x^2-9^2)+x^2. So the answer should be 1-(9!)^2

But sir the polynomial should be x^2(x^2 – 1^2 )(x^2 – 2^2)… (x^2 – 9^2)+x^2.so the answer should be 1-(9!)^2.
( I have not given the exam this year, I read in class 12and will give isi entrance on 2022 though I have solved this in home and got this expression of the polynomial as result. Please inform me if it is wrong

• Ripunjay Dwivedi

sir, in the 4th question, since, g(g(x)) = x, g(x) is a function which is its own inverse. there are three such operations which have these property, which is, they cancel their own effect when acted an even number of times, which are as follows:
(i) the identity function : doing nothing to the input.
(ii) g(x) = -x : multiplication by -1 is cancelled by again multiplying it by -1.
and (iii) taking the reciprocal, i.e., g(x) = 1/x or a/x.
since, g(x) is defined from (o, infinity) to (o, infinity) and is not the identity function, it is the reciprocal i.e., g(x) = a/ x , for some positive, a.
so its derivative is -a/x^2 which is always negative as x^2 and a are always positive. so, the function g(x) is strictly decreasing.

is this correct, sir?

• But sir the polynomial should be x^2(x^2 - 1^2)(x^2-2^2)...(x^2-9^2)+x^2. So the answer should be 1-(9!)^2

But sir the polynomial should be x^2(x^2 – 1^2 )(x^2 – 2^2)… (x^2 – 9^2)+x^2.so the answer should be 1-(9!)^2.
( I have not given the exam this year, I read in class 12and will give isi entrance on 2022 though I have solved this in home and got this expression of the polynomial as result. Please inform me if it is wrong

• IR08 Milan Paul

Sir for problem 5, if we do not write the final answer(-362879) how much mark will be deduced?

• khushal

In question number 5,what concept did you use and it from what book,i have read enough but this was out of the blue.